This arises as a part of my work. Given a positive number $t$, two hermitian matrices $P_1$ and $P_2$, I am interested in knowing if a unit norm vector $z$ exists such that \begin{align} z^{H}P_1z\geq t \\ z^{H}P_2z\geq t \end{align}
A possible reformulation
The vector being unit norm, one can reformulate the above problem as to check whether a unit norm solution exists for \begin{align} z^{H}(P_1-tI)z\geq 0 \\ z^{H}P_2-tI)z\geq 0 \end{align} Note that $t$ is a given positive constant.
HINT: expand $z$ in terms of the eigenvectors of $P_1$ or $P_2$, it's easy to show that the necessary condition is $\min(\lambda_1,\lambda_2) \ge t$, where $\lambda_1$ and $\lambda_2$ are the largest eigenvalue of $P_1$ and $P_2$, respectively.
And if the eigenvector corresponding to $\lambda_1$ is also the eigenvector associated with $\lambda_2$, then this condition is also sufficient.